How many lines of symmetry does a regular octagon have? I can do this problem by drawing a picture and lines of symmetry. My question about this problem is what if it is not an octagon, but any regular polygon. What is a simple way to solve the problem?
Problem: How many lines of symmetry does a regular octagon have?
 A: I'd say there is 8 .
Drawing an octagon is best but if you want to do without it , imagine drawing symmetric lines inbetween the lines of the Octagon or you can imagine drawing lines at the point where of the 2 lines meet.
Lines drawn inbetween lines = 4
Lines drawn where 2 points meet = 4
Total = 8
A: For n-gons there is always $n$ lines of symmetry. Draw different polygons and you will see.
A: For n-gons there are always $2n$ symmetries in total; $n$ reflections and $n$ rotations. So in this case there are 16 symmetries in total, 8 reflections and 8 rotations. 
A nice way to think about this is to consider where you can put each vertex. A symmetry is any permutation that preserves adjacency of vertices. Label the vertices 1 through to 8, then you have 8 choices for where to put the first vertex, 2 for the next and only 1 after that. Thus we have 16 symmetries. 
If you want more information on this look up Dihedral groups.
A: My answer I got was 10 because if you draw lines threw the octagon because it will explain more to you.
             - forth grader advice 
